The present invention relates to an array antenna, particularly relates to an array antenna capable of forming a beampattern with an adjustable beamwidth and low sidelobes.
Conventional Dolph-Tchebysheff arrays were proposed by Dolph in 1946 [1] and are designed by mapping the Tchebysheff polynomial into the array""s space factor. Dolph has proven that for a desired sidelobe level, the Tchebysheff polynomial of order Lxe2x88x921 can be mapped into the spatial factor of a uniform linear array (ULA)xe2x80x94an array of a plurality of antenna elements having a certain inter-element spacexe2x80x94of L elements resulting in a pattern with the sidelobe level as desired and a mainlobe with the minimum possible width. Originally, the design of Dolph-Tchebysheff current distributions was restricted to linear arrays and was applicable to broadside steering only. In fact, little has changed since the classical Dolph-Tchebysheff method came about. For instance, some advances on how to compute the Dolph-Tchebysheff current distribution [1] itself were made by Stegen [3], Davidson [4] and Jazi [5]. More recently, Jazi proposed [6] to use a Tchebysheff polynomial elevated to the n-th so to elevate the order of nulls in the pattern and reduce the number of sidelobes, yielding patterns with the same prescribed sidelobe level, but with a higher directivity, at the expense of slightly broadening the mainlobe.
Finally, the application of Tchebysheff current distributions to a uniform circular array (UCA) was verified [2], making use of a technique that allows the transformation of a UCA into a virtual ULA, long ago presented in [7]. Consequently, a 360xc2x0 spatial span can be scanned with a nearly perfectly invariant beampattern by electronically rotating of a Dolph-Tchebysheff beampattern.
Problem to be Solved by the Invention
The drawback of this approach is that the mainlobe""s beamwidth of the UCA is larger than that of a ULA with the same number of elements and the same inter-element spacing, because the later has a larger aperture than the former. In fact, it is easy to verify from equations (1) and (2) below, which give respectively the broadside aperture of a ULA and the maximum aperture of a UCA with L elements and inter-element spacing xcex94e, that for large arrays the ratio, AULA/AULC=xcfx80, independently of xcex94e.
Equation 1
AULA(L,xcex94e)=(Lxe2x88x921)xcex94exe2x80x83xe2x80x83(1)
                    Equation        ⁢                  xe2x80x83                ⁢        2                            xe2x80x83                                                      A            UCA                    ⁡                      (                          L              ,                              Δ                ⁢                                  xe2x80x83                                ⁢                e                                      )                          =                                            Δ              ⁢                              xe2x80x83                            ⁢              e                                      2              ⁢                              xe2x80x83                            ⁢                              sin                ⁡                                  (                                      π                    L                                    )                                                              ⁢                      (                          1              -                              cos                ⁡                                  (                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        π                                            L                                        ⁢                                          ⌊                                              L                        2                                            ⌋                                                        )                                                      )                                              (        2        )            
Therefore, in applications such as direction of arrival (DOA) estimation in an intelligent transportation system (ITS) where it is desired to perform a spatial scanning only over a limited angular span with a fixed beamwidth, rotation-invariant, equiripple low sidelobe pattern, a ULA would provides the narrowest beam for the same desired SideLobe Ratio (SLR) at broadside. On the other hand, in this case, the mainlobe""s width gets larger as the array is steered to angles closer to end-fire, so that the rotation invariance is lost. Thus, the designer faces the dilemma of either sacrificing too much on the beamwidth by choosing a Dolph-Tchebysheff UCA beampattern or too much on the rotation invariance, by choosing a Dolph-Tchebysheff ULA beampattern.
Yet from another point of view, even when the spatial scanning is desired over the whole 360xc2x0 spatial span, unlike radar applications where the beamwidth of the rotation-invariant array pattern must be as narrow as possible, in communications, there are many applications such as spatial equalization and beam space-time coding, where it is desirable that the beamwidth of the rotation-invariant pattern be adjustable. Therefore, although the contribution in [2] really adds to the flexibility of the design of low sidelobe arrays in terms of its rotation invariance, due to the use of the classical Dolph-Tchebysheff method, it does not deliver the necessary complete flexibility desired.
In order to strengthen the points made above, a brief review of the existing techniques mentioned will be made. We start with the Tchebysheff polynomial used in the classic Dolph design that can be written as below [8].                     Equation        ⁢                  xe2x80x83                ⁢        3                            xe2x80x83                                          T          ⁡                      (                          N              ,              x                        )                          =                  {                                                                      cos                  ⁡                                      (                                          N                      ⁢                                              xe2x80x83                                            ⁢                                              arccos                        ⁡                                                  (                          x                          )                                                                                      )                                                                                                                    if                    ⁢                                          xe2x80x83                                        ⁢                                          "LeftBracketingBar"                      x                      "RightBracketingBar"                                                        ≤                  1                                                                                                      cosh                  ⁡                                      (                                          N                      ⁢                                              xe2x80x83                                            ⁢                      arccos                      ⁢                                              xe2x80x83                                            ⁢                                              h                        ⁡                                                  (                          x                          )                                                                                      )                                                                                                                    if                    ⁢                                          xe2x80x83                                        ⁢                                          "LeftBracketingBar"                      x                      "RightBracketingBar"                                                        ≥                  1                                                                                        (        3        )            
Given a prescribed sidelobe ratio in dB (SLRdB), the voltage sidelobe ratio (SLRV) can be computed according to the following equation:                     Equation        ⁢                  xe2x80x83                ⁢        4                            xe2x80x83                                SLRv        =                              10                                          -                                  SLR                  dB                                            20                                .                                    (        4        )            
In order to design a sidelobe pattern with the given SLRv, first the value of x that makes the |T(N,x)| equal to SLRv is computed. Such a value is given by the following equation:                     Equation        ⁢                  xe2x80x83                ⁢        5                            xe2x80x83                                          x          0                =                  cos          ⁡                      (                                          arc                ⁢                                  xe2x80x83                                ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  h                  ⁡                                      (                    SLRv                    )                                                              N                        )                                              (        5        )            
Note that N=Lxe2x88x921, where L is the element number. Since the Tchebysheff polynomial has only real coefficients and all of its roots lie in the interval x E[xe2x88x92l, l], |T(N,x)| is monotonically increasing for |x| greater than l. Therefore:
Equation 6
|T(N,y)| greater than |T(N,z)| greater than 1∀y greater than z greater than 1xe2x80x83xe2x80x83(6)
Within the interval x E[xe2x88x92l, l], however, the polynomial has its amplitude limited to 1, as can be seen in FIG. 1. Dolph visualized that, if a ULA with L elements and inter-element spacing xcex94e is used, the excitation of the n-th antenna element is calculated by:                     Equation        ⁢                  xe2x80x83                ⁢        7                            xe2x80x83                                          A          n                =                              ∑                          m              =              1                        L                    ⁢                                    T              ⁡                              (                                                      L                    -                    1                                    ,                                                            x                      0                                        ⁢                                          cos                      ⁡                                              (                                                                              π                            ⁢                                                          xe2x80x83                                                        ⁢                            m                                                    L                                                )                                                                                            )                                      ⁢                          ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                                  (                                                            2                      ⁢                      n                                        -                    L                    -                    1                                    )                                ⁢                                                      π                    ⁢                                          xe2x80x83                                        ⁢                    m                                    L                                                                                        (        7        )            
where n=1, 2, . . . , L. If the phases of the signals at all elements are driven so to steer the mainlobe""s peak towards an angle xcex8S, the resulting beampattern will exhibit a space factor exactly given by the equation below (FIG. 2).
Equation 8
|T(Lxe2x88x921,x0 cos(xcfx80xcex94e(cos(xcex8)xe2x88x92cos(xcex8s))))|xe2x80x83xe2x80x83(8)
While the amplitude limitation of the Tchebysheff polynomial is responsible for the equiripple sidelobes, the monotonic behavior for |x| greater than 1 is responsible for its mainlobe""s width inflexibility.
The beamwidth of a pattern steered to xcex8S, can be computed once the criterion that defines the limits of the mainlobe is chosen. In the case of equiripple low sidelobe patterns, a reasonable choice is the point where the mainlobe crosses the sidelobe upper bound, i.e., the sidelobe level beamwidth (xcex94xcex8SL) is defined in terms of the distances between the steering angle (xcex8S) and the angles to the right (xcex8R) and to the left (xcex8L) of the mainlobe""s peak where the gain equals the sidelobe level. If a ULA is used, steering towards any direction rather than broadside causes the mainlobe to enlarge, especially at angles close to the end-fire. Therefore there is a limiting angle after which the beamwidth will enlarge enough to have part of it falling outside the visible region. Making use of the symmetry of the array, this limit can be defined in terms of a minimum steering angle, for which xcex8L vanishes. Thus:                     Equation        ⁢                  xe2x80x83                ⁢        9                            xe2x80x83                                          θ                      s            min                    xe2x80x2                =                              arccos            ⁡                          (                              1                -                                                      1                                          π                      ⁢                                              xe2x80x83                                            ⁢                      Δ                      ⁢                                              xe2x80x83                                            ⁢                      e                                                        ⁢                                      arccos                    ⁡                                          (                                              1                                                  x                          peak                                                                    )                                                                                  )                                .                                    (        9        )            
Equation 10
xcex8xe2x80x2s=arcsin(sin(xcex8s))xe2x80x83xe2x80x83(10)
In the case of the Dolph-Tchebysheff, in the equation above, as well as in those to follow, xpeak, assumes the values of x0 given in equation (5). Then, if xcex8xe2x80x2S greater than xcex8xe2x80x2Smin, xcex8R and xcex8L can be respectively calculated by:                     Equation        ⁢                  xe2x80x83                ⁢        11                            xe2x80x83                                cosh        (                              (                          L              -              1                        )                    ⁢                      (                          β              -                              ⅇ                                  α                  ⁢                                      "LeftBracketingBar"                                                                  x                        peak                                            ⁢                                              cos                        ⁡                                                  (                                                      π                            ⁢                                                          xe2x80x83                                                        ⁢                            Δ                            ⁢                                                          xe2x80x83                                                        ⁢                                                          e                              ⁡                                                              (                                                                                                      cos                                    ⁡                                                                          (                                      θ                                      )                                                                                                        -                                                                      cos                                    ⁡                                                                          (                                                                              θ                                        s                                                                            )                                                                                                                                      )                                                                                                              )                                                                                      "RightBracketingBar"                                                                        )                          "AutoRightMatch"                            (        11        )                                          xe2x80x83                ⁢                              "AutoLeftMatch"                          xe2x80x83                        ⁢                          arccos              ⁢                              xe2x80x83                            ⁢                              h                ⁡                                  (                                                            x                      peak                                        ⁢                                          cos                      ⁡                                              (                                                  π                          ⁢                                                      xe2x80x83                                                    ⁢                          Δ                          ⁢                                                      xe2x80x83                                                    ⁢                                                      e                            ⁡                                                          (                                                                                                cos                                  ⁡                                                                      (                                    θ                                    )                                                                                                  -                                                                  cos                                  ⁡                                                                      (                                                                          θ                                      s                                                                        )                                                                                                                              )                                                                                                      )                                                                              )                                                      )                    =          1                                    xe2x80x83                                ⇒                              (                          L              -              1                        )                    ⁢                      (                          β              -                              ⅇ                                  α                  ⁢                                      "LeftBracketingBar"                                                                  x                        peak                                            ⁢                                              cos                        ⁡                                                  (                                                      π                            ⁢                                                          xe2x80x83                                                        ⁢                            Δ                            ⁢                                                          xe2x80x83                                                        ⁢                                                          e                              ⁡                                                              (                                                                                                      cos                                    ⁡                                                                          (                                      θ                                      )                                                                                                        -                                                                      cos                                    ⁡                                                                          (                                                                              θ                                        s                                                                            )                                                                                                                                      )                                                                                                              )                                                                                      "RightBracketingBar"                                                                        )                                              xe2x80x83                                          xe2x80x83                ⁢                              arccos            ⁢                          xe2x80x83                        ⁢            h            ⁢                          (                                                x                  peak                                ⁢                                  cos                  ⁡                                      (                                          π                      ⁢                                              xe2x80x83                                            ⁢                      Δ                      ⁢                                              xe2x80x83                                            ⁢                                              e                        ⁡                                                  (                                                                                    cos                              ⁡                                                              (                                θ                                )                                                                                      -                                                          cos                              ⁡                                                              (                                                                  θ                                  s                                                                )                                                                                                              )                                                                                      )                                                              )                                =          0                                    xe2x80x83                                          ⇒                                    x              peak                        ⁢                          cos              ⁡                              (                                  π                  ⁢                                      xe2x80x83                                    ⁢                  Δ                  ⁢                                      xe2x80x83                                    ⁢                                      e                    ⁡                                          (                                                                        cos                          ⁡                                                      (                            θ                            )                                                                          -                                                  cos                          ⁡                                                      (                                                          θ                              s                                                        )                                                                                              )                                                                      )                                                    =        1                            xe2x80x83                                          ⇒                      cos            ⁡                          (              θ              )                                      =                                            1                              π                ⁢                                  xe2x80x83                                ⁢                Δ                ⁢                                  xe2x80x83                                ⁢                e                                      ⁢                          acos              ⁡                              (                                  1                                      x                    peak                                                  )                                              +                      cos            ⁡                          (                              θ                s                            )                                                          xe2x80x83            
Using this result, it is obvious that:                     Equation        ⁢                  xe2x80x83                ⁢        12                            xe2x80x83                                                                    θ              L                        =                          arccos              ⁡                              (                                                                            1                                              π                        ⁢                                                  xe2x80x83                                                ⁢                        Δ                        ⁢                                                  xe2x80x83                                                ⁢                        e                                                              ⁢                                          arccos                      ⁡                                              (                                                  1                                                      x                            peak                                                                          )                                                                              +                                      cos                    ⁡                                          (                                              θ                        s                                            )                                                                      )                                              ⁢                      
                    ⁢                                    if              ⁢                              xe2x80x83                            ⁢                              arcsin                ⁡                                  (                                      sin                    ⁡                                          (                                              θ                        s                                            )                                                        )                                                       less than                           θ                              s                min                                                    ⁢                  
                                    (        12        )                                Equation        ⁢                  xe2x80x83                ⁢        13                            xe2x80x83                                                      θ            R                    =                      π            -                          arccos              ⁡                              (                                                                            1                                              π                        ⁢                                                  xe2x80x83                                                ⁢                        Δ                        ⁢                                                  xe2x80x83                                                ⁢                        e                                                              ⁢                                          arccos                      ⁡                                              (                                                  1                                                      x                            peak                                                                          )                                                                              -                                      cos                    ⁡                                          (                                              θ                        s                                            )                                                                      )                                                    ⁢                  
                ⁢                              if            ⁢                          xe2x80x83                        ⁢                          arcsin              ⁡                              (                                  sin                  ⁡                                      (                                          θ                      s                                        )                                                  )                                               less than                       θ                          s              min                                                          (        13        )            
Outside the limits, it is easy to see that xcex8R and xcex8L can be respectively calculated by the following equation:                     Equation        ⁢                  xe2x80x83                ⁢        14                            xe2x80x83                                                                                    cosh                (                                                      (                                          L                      -                      1                                        )                                    ⁢                                      (                                          β                      -                                              ⅇ                                                  α                          ⁢                                                      "LeftBracketingBar"                                                                                          x                                peak                                                            ⁢                                                              cos                                ⁡                                                                  (                                                                      π                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    Δ                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          e                                      ⁡                                                                              (                                                                                                                              cos                                            ⁡                                                                                          (                                              θ                                              )                                                                                                                                -                                                                                      cos                                            ⁡                                                                                          (                                                                                              θ                                                s                                                                                            )                                                                                                                                                                      )                                                                                                                                              )                                                                                                                      "RightBracketingBar"                                                                                                                )                                    ⁢                  arccos                  ⁢                                      xe2x80x83                                    ⁢                                      h                    ⁡                                          (                                                                        x                          peak                                                ⁢                                                  cos                          ⁡                                                      (                                                          π                              ⁢                                                              xe2x80x83                                                            ⁢                              Δ                              ⁢                                                              xe2x80x83                                                            ⁢                                                              e                                ⁡                                                                  (                                                                                                            cos                                      ⁡                                                                              (                                        θ                                        )                                                                                                              -                                                                          cos                                      ⁡                                                                              (                                                                                  θ                                          s                                                                                )                                                                                                                                              )                                                                                                                      )                                                                                              )                                                                      )                            =              1                                                                                          ⇒                                                      (                                          L                      -                      1                                        )                                    ⁢                                      (                                          β                      -                                              ⅇ                                                  α                          ⁢                                                      "LeftBracketingBar"                                                                                          x                                peak                                                            ⁢                                                              cos                                ⁡                                                                  (                                                                      π                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    Δ                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          e                                      ⁡                                                                              (                                                                                                                              cos                                            ⁡                                                                                          (                                              θ                                              )                                                                                                                                -                                                                                      cos                                            ⁡                                                                                          (                                                                                              θ                                                s                                                                                            )                                                                                                                                                                      )                                                                                                                                              )                                                                                                                      "RightBracketingBar"                                                                                                                )                                    ⁢                  arccos                  ⁢                                      xe2x80x83                                    ⁢                                      h                    (                                                                  x                        peak                                            ⁢                                              cos                        ⁡                                                  (                                                      π                            ⁢                                                          xe2x80x83                                                        ⁢                            Δ                            ⁢                                                          xe2x80x83                                                        ⁢                                                          e                              ⁡                                                              (                                                                                                      cos                                    ⁡                                                                          (                                      θ                                      )                                                                                                        -                                                                      cos                                    ⁡                                                                          (                                                                              θ                                        s                                                                            )                                                                                                                                      )                                                                                                              )                                                                                      )                                                              =                                                (                                      L                    -                    1                                    )                                ⁢                π                                                                                                        ⇒                                                      x                    peak                                    ⁢                                      cos                    ⁡                                          (                                              π                        ⁢                                                  xe2x80x83                                                ⁢                        Δ                        ⁢                                                  xe2x80x83                                                ⁢                                                  e                          ⁡                                                      (                                                                                          cos                                ⁡                                                                  (                                  θ                                  )                                                                                            -                                                              cos                                ⁡                                                                  (                                                                      θ                                    s                                                                    )                                                                                                                      )                                                                                              )                                                                                  =              π                                                                                          ⇒                                  cos                  ⁡                                      (                    θ                    )                                                              =                                                                    π                                          π                      ⁢                                              xe2x80x83                                            ⁢                      Δ                      ⁢                                              xe2x80x83                                            ⁢                      e                                                        ⁢                                      acos                    ⁡                                          (                                                                        -                          1                                                                          x                          peak                                                                    )                                                                      +                                  cos                  ⁡                                      (                                          θ                      s                                        )                                                                                                                          xe2x80x83                                                          (        14        )                                Equation        ⁢                  xe2x80x83                ⁢        15                            xe2x80x83                                                      θ            L                    =                                    π              -                                                arccos                  ⁡                                      (                                                                                            1                                                      π                            ⁢                                                          xe2x80x83                                                        ⁢                            Δ                            ⁢                                                          xe2x80x83                                                        ⁢                            e                                                                          ⁢                                                  arccos                          ⁡                                                      (                                                                                          -                                1                                                                                            x                                peak                                                                                      )                                                                                              -                                              cos                        ⁡                                                  (                                                      θ                            s                                                    )                                                                                      )                                                  ⁢                                  xe2x80x83                                ⁢                if                ⁢                                  xe2x80x83                                ⁢                                  arcsin                  ⁡                                      (                                          sin                      ⁡                                              (                                                  θ                          s                                                )                                                              )                                                                         less than                           θ                              s                min                                                    ⁢                  
                                    (        15        )                                Equation        ⁢                  xe2x80x83                ⁢        16                            xe2x80x83                                          θ          R                =                                            arccos              ⁡                              (                                                                            1                                              π                        ⁢                                                  xe2x80x83                                                ⁢                        Δ                        ⁢                                                  xe2x80x83                                                ⁢                        e                                                              ⁢                                          arccos                      ⁡                                              (                                                                              -                            1                                                                                x                            peak                                                                          )                                                                              +                                      cos                    ⁡                                          (                                              θ                        s                                            )                                                                      )                                      ⁢                          xe2x80x83                        ⁢            if            ⁢                          xe2x80x83                        ⁢                          arcsin              ⁡                              (                                  sin                  ⁡                                      (                                          θ                      s                                        )                                                  )                                               less than                       θ                          s              min                                                          (        16        )            
With the above equations in hand, xcex94xcex8SL can finally be computed by:                     Equation        ⁢                  xe2x80x83                ⁢        17                            xe2x80x83                                          Δ          ⁢                      xe2x80x83                    ⁢                      θ            SL                          =                  {                                                                                          θ                    R                                    -                                      θ                    L                                                                                                                    if                    ⁢                                          xe2x80x83                                        ⁢                                          θ                      s                      xe2x80x2                                                        ≥                                      θ                                          s                      min                                        xe2x80x2                                                                                                                        π                  -                                      θ                    R                                    +                                      θ                    L                                                                                                                    if                    ⁢                                          xe2x80x83                                        ⁢                                          θ                      s                      xe2x80x2                                                         less than                                       θ                                          s                      min                                        xe2x80x2                                                                                                          (        17        )            
Since in Dolph-Tchebysheff arrays x0 is a function of the desired sidelobe ratio and the number of elements as given in equation (5), the above formulas yield the direct relationship between the steering direction and the beamwidth. In other words, in the classic design, the beamwidth is a function of the steering direction xcex8S, the number of elements in the array L, its inter-element spacing xcex94e and the desired sidelobe ratio SLRv. Moreover, it was shown in [1] that, for fixed xcex8S, L, xcex94e and SLRv, the beamwidth of the Dolph-Tchebysheff array is the minimum possible. FIG. 3 shows the xe2x88x9240 dB Dolph-Tchebysheff patterns of a ULA with 20 elements. It can be seen how the mainlobe""s shape change (gets distorted) as it is steered to angles closer to the end-fire.
Next, consider the application of conventional Dolph-Tchebysheff to a UCA, as proposed in [2]. For convenience, a brief review of the necessary formulas and procedures is given below. Be the steering vector of a uniform circular array of omni-directional elements and minimum inter-element distance equal to xcex94e be:       Equation    ⁢          xe2x80x83        ⁢    18                                            a            ⁡                          (              θ              )                                =                                    [                              ⅇ                                                      -                    j                                    ⁢                                                            2                      ⁢                      π                      ⁢                                              xe2x80x83                                            ⁢                      R                                        λ                                    ⁢                                      cos                    (                                                                                            2                          ⁢                                                      π                            ⁡                                                          (                                                              i                                -                                1                                                            )                                                                                                      L                                            ⁢                      θ                                                                                  ]                                      i              =                                                1                  ,                                      xe2x80x83                                    ⁢                  …                  ⁢                                      xe2x80x83                                    ,                                      xe2x80x83                                    ⁢                  L                                -                1                                                                          (          18          )                    
where R=xcex94excex/2 sin(xcfx80/L). Define the matrix F by:       Equation    ⁢          xe2x80x83        ⁢    19                                                          F              =                                                1                                      L                                                  ⁡                                  [                                                                                    1                                                                                              ω                                                      -                            h                                                                                                                      ⋯                                                                                              ω                                                                                    -                                                              (                                                                  L                                  -                                  1                                                                )                                                                                      ⁢                            h                                                                                                                                                              ⋮                                                                    ⋮                                                                    ⋯                                                                    ⋮                                                                                                            1                                                                                              ω                                                      -                            1                                                                                                                      ⋯                                                                                              ω                                                      -                                                          (                                                              L                                -                                1                                                            )                                                                                                                                                                                          1                                                                    1                                                                    …                                                                    1                                                                                                            1                                                                                              ω                          1                                                                                            ⋯                                                                                              ω                                                      (                                                          L                              -                              1                                                        )                                                                                                                                                              ⋮                                                                    ⋮                                                                    ⋯                                                                    ⋮                                                                                                            1                                                                                              ω                          h                                                                                            ⋯                                                                                              ω                                                                                    (                                                              L                                -                                1                                                            )                                                        ⁢                            h                                                                                                                                ]                                                      ⁢                          
                        ⁢                                          where                ⁢                                  xe2x80x83                                ⁢                ω                            =                                                ⅇ                                      j2π                    /                    L                                                  ⁢                                  xe2x80x83                                ⁢                and                                      ⁢                          
                        ⁢                          Equation              ⁢                              xe2x80x83                            ⁢              20                                ⁢                      
                                                (          19          )                                              h          =                      max            ⁢                          {                                                h                  ≤                                                            L                      -                      1                                        2                                                  |                                                                            |                                                                        J                                                      h                            -                            L                                                                          ⁡                                                  (                                                                                    2                              ⁢                              π                              ⁢                                                              xe2x80x83                                                            ⁢                              R                                                        λ                                                    )                                                                    |                                                              |                                                                        J                          h                                                ⁡                                                  (                                                                                    2                              ⁢                              π                              ⁢                                                              xe2x80x83                                                            ⁢                              R                                                        λ                                                    )                                                                    |                                                        ≤                                      ϵ                    ⁢                                           less than  less than                       1                                                                                  }                                                            (          20          )                    
Define the matrix J by:       Equation    ⁢          xe2x80x83        ⁢    21                                J          =                      diag            ⁢                          {                                                (                                                            j                      m                                        ⁢                                          L                                        ⁢                                                                  J                        m                                            ⁡                                              (                                                                              2                            ⁢                            π                            ⁢                                                          xe2x80x83                                                        ⁢                            R                                                    λ                                                )                                                                              )                                                  -                  1                                            }                                                            (          21          )                    
with m=xe2x88x92h, . . . , o, . . . , h and where Jxe2x88x92m(x)=(xe2x88x921)mJm(x). Then, if right-multiply JF to a(xcex8), the following can be obtained.
Equation 22
A(xcex8)=JF{haeck over (a)}(xcex8)=[exe2x88x92jhxcex8 . . . 1 . . . ejhxcex8].xe2x80x83xe2x80x83(22)
Observe that the resulting steering vector is similar to the steering vector of a linear array, except that now the elements of A(xcex8) are no longer dependent on cos(xcex8), on xcex8directly.
Using of this transformation the design of Dolph-Tchebysheff current distributions for a UCA is possible, and the beamwidth of this pattern is not dependent on xcex8. It was shown in [2] that, in this case, equation (17) reduces to:       Equation    ⁢          xe2x80x83        ⁢    23                                            Δθ            SL                    =                                                    θ                R                            -                              θ                L                                      ⩵                          4              ⁢                              xe2x80x83                            ⁢                              arccos                ⁡                                  (                                      1                                          x                      peak                                                        )                                                                                          (          23          )                    
Once the virtual ULA transformed from an L-elements UCA has 2h+l elements as shown in [2], equation (5) reduces to       Equation    ⁢          xe2x80x83        ⁢    24                                            x            peak                    =                                    x              0                        =                                          cosh                ⁡                                  (                                                            arccosh                      ⁡                                              (                                                  SLR                          v                                                )                                                                                    2                      ⁢                      h                                                        )                                            ⁢                              xe2x80x83                            .                                                            (          24          )                    
Therefore for the classic Dolph-Tchebysheff UCA, it gives:       Equation    ⁢          xe2x80x83        ⁢    25                                            Δθ            SL                    =                                                    θ                R                            -                              θ                L                                      =                          4              ⁢                              xe2x80x83                            ⁢                                                arccos                  ⁡                                      (                                          1                                              cosh                        ⁡                                                  (                                                                                    arccos                              ⁢                                                              xe2x80x83                                                            ⁢                                                              h                                ⁡                                                                  (                                  SLRv                                  )                                                                                                                                                    2                              ⁢                              h                                                                                )                                                                                      )                                                  ⁢                                  xe2x80x83                                .                                                                          (          25          )                    
The above expression clearly shows, as previously stated, that in the scheme proposed in [2], the beamwidth of the low sidelobe pattern independent of xcex8S, being a function of the number of elements, the inter-element spacing and the prescribed sidelobe ratio. FIG. 4 illustrates the result delivered by the technique proposed in [2], showing the beampatterns obtained with a UCA of 35 elements and half wavelength inter-element spacing (what gives h=15), steered to 0xc2x0 and 60xc2x0. As expected, it can be seen that there is no distortion caused by steering.
The drawback of such design, however, is that since a UCA has an aperture narrower than that of a ULA with the same number of elements and the same inter-element spacing, the beamwidth of the beamformer proposed in [2], that is, xcex94xcex8SL given in equation (25), is significantly larger than that of the Dolph-Tchebysheff beampattern given in equation (17), with a ULA not only at broadside exactly, but within a significantly large angular span around it. FIG. 5 clarifies this fact by showing the curves of beamwidth against steering direction of ULA and UCA Dolph-Tchebysheff beampatterns with L=20 and xcex94e=0.5xcex for prescribed xe2x88x9220 dB and xe2x88x9240 dB sidelobe ratios. It is seen that the price paid for a rotation-invariant mainlobe obtained with a UCA is the significant enlargement of the beamwidth. From all the explained in this section it is possible to conclude the following about conventional Dolph-Tchebysheff array design:
1) There is no way to adjust the mainlobe""s width to a desired value (beyond a practical minimum) without interfering on the sidelobe.
2) If ULA is used, given L, SLR and xcex94e, the mainlobe""s width varies as a function of the steering direction.
3) If UCA is used, the mainlobe""s width is much larger than that of a ULA with the same L, SLR and xcex94e.
Following is a list of the literature mentioned above:
[1] C. L. Dolph: xe2x80x9cA Current Distribution for Broadside Arrays Which Optimizes the Relationship Between Beamwidth and Sidelobe Levelxe2x80x9d, Proc. IRE, 34(6), pp. 335-348, 1946.
[2] B. K. Lau and Y. H. Leung: xe2x80x9cA Dolph-Chebyshev Approach to the Synthesis of Array Patterns for Uniform Circular Arraysxe2x80x9d, Proc. ISCAS 2000, vol. I, pp. 124-127, May 2000.
[3] R. J. Stegen: xe2x80x9cExcitation Coefficients and Beamwidths of Tchebysheff Arraysxe2x80x9d, Proc. IRE, 40(11), pp.1671-1674, 1953.
[4] T. N. Davidson: xe2x80x9cA Note on the Calculation of Dolph-Chebyshev Shading for Linear Arrayxe2x80x9d, ASPL-1991-3, Dept. Electrical and Electronic Eng., the University of Western Australia, Aug, 1991.
[5] S. Jazi: xe2x80x9cA New Formulation for the Design of Chebyshev Arraysxe2x80x9d, IEEE Trans. Antennas and Propagations, 42(3), pp.439-443, 1994.
[6] S. Jazi: xe2x80x9cModified Chebyshev Arraysxe2x80x9d, IEE Proc. on Microwaves, Antennas and Propagations, vol.145, no.1, February 1998.
[7] D. E. N. Davies: xe2x80x9cA Transformation Between the Phasing Technique Required for Linear and Circular Aerial Arraysxe2x80x9d, Proc. IEE 112(11), pp.2041-2045, 1965.
[8] Y. Y. Lo and S. W. Lee: xe2x80x9cAntenna Handbook. Theory, Applications and Designxe2x80x9d, VNR, 1988.
The present invention was made in consideration with the above circumstances and has as an objective thereof to provide an array antenna capable of forming a beampattern with an adjustable beamwidth and low sidelobes ratio.
Means for Solving the Problem
To achieve the above object, according to the present invention, there is provided an array antenna comprising a plurality of antenna elements, a calculation means for calculating excitation coefficients for each said antenna element in a way such that said antenna elements form a beampattern having a flat top mainlobe of adjustable beamwidth and a predetermined sidelobe level.
Further, according to the present invention, there is provided an array antenna comprising a plurality of antenna elements, a calculation means for calculating excitation coefficients An for each said antenna element in accordance with                               A          n                =                              ∑                          m              =              1                        L                    ⁢                      xe2x80x83                    ⁢                                    G              ⁡                              (                                  L                  -                                      1                    ,                                          xe2x80x83                                        ⁢                                          x                      p                                        ⁢                                          cos                      ⁡                                              (                                                                              π                            ⁢                                                          xe2x80x83                                                        ⁢                            m                                                    L                                                )                                                              ,                                          xe2x80x83                                        ⁢                    α                    ,                                          xe2x80x83                                        ⁢                    β                                                  )                                      ⁢                          ⅇ                                                -                                      j                    ⁡                                          (                                                                        2                          ⁢                          n                                                -                        L                        -                        1                                            )                                                                      ⁢                                                      π                    ⁢                                          xe2x80x83                                        ⁢                    m                                    L                                                                                        (        36        )            
where the function G(N,x,xcex1,xcex2) is given as                               G          ⁡                      (                          N              ,                              xe2x80x83                            ⁢              x              ,                              xe2x80x83                            ⁢              α              ,                              xe2x80x83                            ⁢              β                        )                          =                  {                                                                                          cos                    ⁡                                          (                                                                        N                          ⁡                                                      (                                                          β                              -                                                              ⅇ                                                                  α                                  |                                  x                                  |                                                                                                                      )                                                                          ⁢                                                  arccos                          ⁡                                                      (                            x                            )                                                                                              )                                                                                                                                        cosh                    ⁡                                          (                                                                        N                          ⁡                                                      (                                                          β                              -                                                              ⅇ                                                                  α                                  |                                  x                                  |                                                                                                                      )                                                                          ⁢                        arccos                        ⁢                                                  xe2x80x83                                                ⁢                                                  h                          ⁡                                                      (                            x                            )                                                                                              )                                                                                            ⁢                                                                                                                              if                        |                        x                        |                                                  ≤                          1                                                                    ⁢                                              xe2x80x83                                                                                                                                                        if                      |                      x                      |                                              ≥                        1                                                                                                        .                                                          (        26        )            
Preferably, according to the present invention, said antenna elements form a uniform linear array.
Preferably, according to the present invention, said antenna elements form a uniform circular array.